Average Error: 29.9 → 7.8
Time: 20.1s
Precision: 64
\[0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}
double f(double a, double b) {
        double r2553944 = a;
        double r2553945 = r2553944 * r2553944;
        double r2553946 = b;
        double r2553947 = r2553946 * r2553946;
        double r2553948 = r2553945 - r2553947;
        double r2553949 = r2553948 / r2553945;
        double r2553950 = fabs(r2553949);
        double r2553951 = sqrt(r2553950);
        return r2553951;
}
↓
double f(double a, double b) {
        double r2553952 = 1.0;
        double r2553953 = b;
        double r2553954 = a;
        double r2553955 = r2553953 / r2553954;
        double r2553956 = r2553955 * r2553955;
        double r2553957 = r2553952 - r2553956;
        double r2553958 = fabs(r2553957);
        double r2553959 = sqrt(r2553958);
        return r2553959;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.9

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]

  2. Simplified7.8

    \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}}\]

  3. Final simplification7.8

    \[\leadsto \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))